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((2 - 2*I)*(-1)^(1/4)*ArcTanh[(1 + Tan[x/4])/Sqrt[2]]*(Cos[x/2] - Sin[x/2]))/Sqrt[1 - Sin[x]]


Integrate[1/Sqrt[1 - Sin[x]], x] ==
((2 - 2*I)*(-1)^(1/4)*ArcTanh[(1 + Tan[x/4])/Sqrt[2]]* (Cos[x/2] - Sin[x/2]))/Sqrt[1 - Sin[x]]



                 1
Integrate[----------------, x] == 
          Sqrt[1 - Sin[x]]

                                   x
                           1 + Tan[-]
               1/4                 4
((2 - 2 I) (-1)    ArcTanh[----------] 
                            Sqrt[2]
 
         x        x
    (Cos[-] - Sin[-])) / Sqrt[1 - Sin[x]]
         2        2

Time to compute: 0.09 second

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